The reduction also implies improved regret bounds for misspecified contextual linear bandits by leveraging the optimal and nearly optimal regret bounds proved in the literature [1]} for fixed action sets. This results in high probability bounds; as opposed to the best known results for changing action sets that add \(\log \)  factors and only hold in expectation. As the reduction introduces a small \(\tilde{O}(1/\sqrt{T})\)  misspecification, this adds to the possibly unknown misspecification \(\epsilon \) .
There is a little subtlety here; some algorithms adapt to known \(\epsilon \)  better than the unknown case. To avoid adding extra factors to the regret bounds by sub-optimal adaptation to the known \(\tilde{O}(1/\sqrt{T})\)  part of misspecification, we slightly modify existing algorithms to account for this. This is done by increasing the number of times an arm is explored by a constant factor.
Similarly, our results allow to carry over the better regret bounds for linear bandits with adversarial corruption to the contextual setting. This is achieved by modifying the algorithms for linear bandits with adversarial corruption to account for the known misspecification added by our reduction.
